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    <assign|math|<macro|x|<math-colored|<with|math-display|true|<uncolored-math|<arg|x>>>>>>
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  <screens|<\shown>
    <chapter*|Chapter 2: Limits and Derivatives>

    <section|The Tangent and Velocity Problems>

    <subsection|The tangent problem (how to find the tangent line)>

    \;

    <with|gr-mode|<tuple|group-edit|edit-props>|gr-frame|<tuple|scale|1cm|<tuple|0.28001gw|0.499999gh>>|gr-geometry|<tuple|geometry|1par|0.6par>|gr-grid|<tuple|cartesian|<point|0|0>|1>|gr-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-aspect|<tuple|<tuple|axes|none>|<tuple|1|none>|<tuple|10|none>>|gr-edit-grid|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-fill-color|red|gr-color|blue|<graphics||<spline|<point|-2.95384|0.5>|<point|-0.7|-0.7>|<point|2.0|1.7>>|<math-at|y=f<around*|(|x|)>|<point|2.2|1.7>>|<with|color|red|fill-color|red|<point|-0.4|-0.602261>>|<with|color|red|<text-at|secant
    line|<point|1.9|0.4>>>|<with|color|red|<math-at|<around*|(|a,f<around*|(|a|)>|)>|<point|-1.53969986357435|-0.260668072264252>>>|<with|color|red|<math-at|<around*|(|b,f<around*|(|b|)>|)>|<point|0.23470999214519|1.11374178345529>>>|<with|color|blue|fill-color|red|<line|<point|-0.4|-0.602261>|<point|1.6|0.3>>>|<with|color|red|fill-color|red|<line|<point|-0.4|-0.602261>|<point|1.47921610728005|1.0>>>|<with|color|red|<math-at|y=<frac|f<around*|(|b|)>-f<around*|(|a|)>|b-a>*<around*|(|x-a|)>+f<around*|(|a|)>|<point|2.0|-0.3>>>|<with|color|blue|fill-color|red|<\document-at>
      \;
    </document-at|<point|3.2|-2.5>>>|<with|color|blue|<\document-at>
      In fact,

      <\equation*>
        m=lim<rsub|b\<rightarrow\>a> <frac|f<around*|(|b|)>-f<around*|(|a|)>|b-a>
      </equation*>
    </document-at|<point|3.7|-1.3>>>|<with|color|blue|<\document-at>
      tangent line:

      <\equation*>
        y=m*x+f<around*|(|a|)>
      </equation*>

      \ for some slope m
    </document-at|<point|0.3|-1.0>>>|<\with|fill-color|red|color|blue>
      \;
    </with>>>

    <subsection|The velocity problem (how to find instantaneous velocity)>

    <\unfolded-plain>
      Let <math|S<around*|(|t|)>> be the position of an object at time
      <math|t>. The average velocity from <math|t=a> to <math|t=b> is

      <\equation*>
        <frac|S<around*|(|b|)>-S<around*|(|a|)>|b-a>
      </equation*>

      The instantaneous velocity is the limit of the average velocity in
      shorter and shorter time intervals, i.e.

      <\equation*>
        lim<rsub|b\<rightarrow\>a> <frac|S<around*|(|b|)>-S<around*|(|a|)>|b-a>
      </equation*>

      <\remark>
        Limit is always a dynamic process!
      </remark>
    <|unfolded-plain>
      \;
    </unfolded-plain>

    <section|The Limit of a Function>

    <subsection|Intuitive definition of a limit>

    <\definition>
      For a given function <math|f<around*|(|x|)>>, and a point <math|x=a>
      <deleted|in the domain of <math|f>>, we say

      <\equation*>
        <with|color|blue|<with|color|red|lim<rsub|x\<rightarrow\>a>
        f<around*|(|x|)>=L>>,
      </equation*>

      read as: the limit of <math|f<around*|(|x|)>>, as <math|x> approaches
      <math|a>, equals <math|L>, if <math|f<around*|(|x|)>> gets arbitrarily
      close to <math|L> as long as <math|x> is sufficiently close to
      <math|a>.
    </definition>

    <\example>
      Let <math|f<around*|(|x|)>=x<rsup|2>>, then\ 

      <\equation*>
        lim<rsub|x\<rightarrow\>3> f<around*|(|x|)>=9
      </equation*>
    </example>

    <\wide-block>
      <tformat|<cwith|1|1|1|1|cell-lsep|1spc>|<cwith|1|1|1|1|cell-rsep|1spc>|<cwith|1|1|1|1|cell-bsep|1spc>|<cwith|1|1|1|1|cell-tsep|1spc>|<table|<row|<\cell>
        <\example>
          Find the equation of the tangent line to the graph of
          <math|f<around*|(|x|)>=x<rsup|2>> at <math|x=1>.

          <\answer*>
            Draw a secant line from <math|<around*|(|1,1|)>> to
            <math|<around*|(|x,f<around*|(|x|)>|)>>. Its slope is

            <\equation*>
              m<around*|(|x|)>=<frac|f<around*|(|x|)>-1|x-1>=<frac|x<rsup|2>-1|x-1>.
            </equation*>

            Then the slope of the tangent line is

            <\equation*>
              lim<rsub|x\<rightarrow\>1> m<around*|(|x|)>=lim<rsub|x\<rightarrow\>1>
              <frac|x<rsup|2>-1|x-1>=lim<rsub|x\<rightarrow\>1> x+1=2.
            </equation*>

            So the equation of the tangent line is

            <\equation*>
              y=2*<around*|(|x-1|)>+1=2*x-1.
            </equation*>
          </answer*>
        </example>
      </cell>>>>
    </wide-block>

    <\question>
      \;

      <\equation*>
        lim<rsub|x\<rightarrow\>0> <frac|sin x|x>=?
      </equation*>

      Hint: draw a circle.

      <with|gr-mode|<tuple|edit|math-at>|gr-frame|<tuple|scale|1cm|<tuple|0.5gw|0.5gh>>|gr-geometry|<tuple|geometry|1par|0.6par>|gr-grid|<tuple|cartesian|<point|0|0>|2>|gr-grid-old|<tuple|cartesian|<point|0|0>|2>|gr-edit-grid-aspect|<tuple|<tuple|axes|none>|<tuple|1|none>|<tuple|10|none>>|gr-edit-grid|<tuple|cartesian|<point|0|0>|2>|gr-edit-grid-old|<tuple|cartesian|<point|0|0>|2>|gr-snap|<tuple>|gr-arrow-end|\<gtr\>|gr-color|blue|<graphics||<carc|<point|-2|0>|<point|2.0|0.0>|<point|0.0|-2.0>>|<with|color|blue|<line|<point|0|0>|<point|2.0|0.0>>>|<with|color|blue|<line|<point|0|0>|<point|1.73205080756888|1.0>>>|<with|color|blue|<math-at|x|<point|0.6|0.114858>>>|<with|color|blue|<math-at|r=1|<point|0.8|-0.300381>>>|<with|color|blue|<line|<point|1.73428|0.999297>|<point|1.7438009012361|0.00869816858902807>>>|<with|color|blue|arrow-end|\<gtr\>|<line|<point|2.34388|1.1517>|<point|1.7819008640291|0.427797759312084>>>|<with|color|blue|<math-at|sin
      x|<point|2.55343|1.24695>>>|<with|arrow-end|\<gtr\>|<line|<point|2.91538|0.627823>|<point|2.02955062218364|0.294447889536566>>>|<math-at|x|<point|3.09636|0.637348>>|<with|color|blue|<math-at|lim<rsub|x\<rightarrow\>0>
      <frac|sin x|x>=1|<point|2.34388|-0.705676>>>>>
    </question>

    <\example>
      \;

      <\equation*>
        lim<rsub|x\<rightarrow\>0> sin <frac|1|x>=?
      </equation*>

      The limit does not exist!\ 
    </example>

    <\example>
      \;

      <\equation*>
        lim<rsub|x\<rightarrow\>0> x*sin <frac|1|x>=0
      </equation*>
    </example>

    <with|ornament-border|1ln|ornament-color|red|ornament-shape|rounded|<\remark>
      In the definition of <math|lim<rsub|x\<rightarrow\>a>
      f<around*|(|x|)>=L>, the function value <math|f<around*|(|a|)>> is
      irrelevant. In fact, <math|f> can even be undefined on <math|x=a>.
    </remark>>

    <\example>
      Let

      <\equation*>
        f<around*|(|x|)>=<choice|<tformat|<table|<row|<cell|1-x,>|<cell|x\<less\>0,>>|<row|<cell|e<rsup|x>,>|<cell|x\<gtr\>0,>>|<row|<cell|5,>|<cell|x=0,>>>>><space|1em>g<around*|(|x|)>=<choice|<tformat|<table|<row|<cell|1-x,>|<cell|x\<less\>0,>>|<row|<cell|e<rsup|x>,>|<cell|x\<gtr\>0.>>>>>
      </equation*>

      Find <math|lim<rsub|x\<rightarrow\>0> f<around*|(|x|)>> and
      <math|lim<rsub|x\<rightarrow\>0> g<around*|(|x|)>>.

      <\answer*>
        <\equation*>
          lim<rsub|x\<rightarrow\>0> f<around*|(|x|)>=1\<neq\>f<around*|(|0|)>
        </equation*>

        <\equation*>
          lim<rsub|x\<rightarrow\>0> g<around*|(|x|)>=1,<space|1em>g<around*|(|0|)>
          <text|is undefined>
        </equation*>
      </answer*>
    </example>

    <subsection|One-sided limits>

    <\definition>
      For a given function <math|f<around*|(|x|)>>, and a point <math|x=a>,
      \ we say

      <\equation*>
        <with|color|blue|<with|color|red|lim<rsub|x\<rightarrow\><with|color|dark
        green|a<rsup|->>> f<around*|(|x|)>=L>>,
      </equation*>

      read as: the <strong|left-hand limit> of <math|f<around*|(|x|)>>, as
      <math|x> approaches <math|a>, equals <math|L>, if
      <math|f<around*|(|x|)>> gets arbitrarily close to <math|L> as long as
      <math|x\<less\>a> is sufficiently close to <math|a>.

      Definition of the <strong|right-hand limit> is similar.\ 
    </definition>

    <\example>
      Find the left and right hand limits of

      <\equation*>
        f<around*|(|x|)>=<choice|<tformat|<table|<row|<cell|x-1,>|<cell|x\<less\>0,>>|<row|<cell|sin
        x+3,>|<cell|x\<gtr\>0.>>>>>
      </equation*>

      <\answer*>
        <\equation*>
          lim<rsub|x\<rightarrow\>0<rsup|->>
          f<around*|(|x|)>=-1,<space|1em>lim<rsub|x\<rightarrow\>0<rsup|+>>
          f<around*|(|x|)>=3.
        </equation*>
      </answer*>
    </example>

    \;

    Relation between limit and one-sided limits:

    <\itemize>
      <item>If the limit exists, the both left and right hand limit exsits
      and equal to the limit value itself

      <item>If both left and right hand limit exists and equal to the same
      value, then the limit exists and equal to the same value.

      <item>If both left and right hand limit exists but they are not equal,
      then the limit does not exist.\ 
    </itemize>

    \;

    <\example>
      \ 

      <\wide-tabular>
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        </cell>|<\cell>
          <\itemize>
            <item><math|<with|math-display|true|lim<rsub|x\<rightarrow\>2<rsup|->>
            g<around*|(|x|)>=3>>

            <item><math|<with|math-display|true|lim<rsub|x\<rightarrow\>2<rsup|+>>
            g<around*|(|x|)>=1>>

            <item><math|<with|math-display|true|lim<rsub|x\<rightarrow\>2>
            g<around*|(|x|)>>> not exist!

            <item><math|g<around*|(|2|)>> is undefined!

            <item><math|<with|math-display|true|lim<rsub|x\<rightarrow\>5<rsup|->>
            g<around*|(|x|)>=2>>

            <item><math|<with|math-display|true|lim<rsub|x\<rightarrow\>5<rsup|+>>
            g<around*|(|x|)>=2>>

            <item><math|<with|math-display|true|lim<rsub|x\<rightarrow\>5<rsup|>>
            g<around*|(|x|)>=2>>

            <item><math|g<around*|(|5|)>\<approx\>1.2>

            <item><math|<with|math-display|true|lim<rsub|x\<rightarrow\>3>
            g<around*|(|x|)>=1.5=g<around*|(|3|)>>>
          </itemize>
        </cell>>>>
      </wide-tabular>
    </example>

    <subsection|Infinite limits>

    <\definition>
      We write

      <\equation*>
        lim<rsub|x\<rightarrow\>a> f<around*|(|x|)>=\<infty\>
      </equation*>

      if <math|f<around*|(|x|)>> gets arbitrarily large when <math|x> is
      sufficiently close to <math|a>.

      Similarly, we can define

      <\equation*>
        lim<rsub|x\<rightarrow\>a> f<around*|(|x|)>=-\<infty\>
      </equation*>
    </definition>

    <\note*>
      In these cases we will not say the limit exists.\ 
    </note*>

    <\example*>
      <\equation*>
        lim<rsub|x\<rightarrow\>0> <frac|1|x<rsup|2>>=\<infty\>
      </equation*>
    </example*>

    <strong|One-sided infinite limits> can be defined similarly

    <\equation*>
      lim<rsub|x\<rightarrow\>a<rsup|->> f<around*|(|x|)>=\<infty\><infix-or>-\<infty\>
    </equation*>

    <\equation*>
      lim<rsub|x\<rightarrow\>a<rsup|+>> f<around*|(|x|)>=\<infty\><infix-or>-\<infty\>
    </equation*>

    <\example*>
      <\equation*>
        lim<rsub|x\<rightarrow\>0<rsup|+>>
        <frac|1|x>=\<infty\>,<space|1em>lim<rsub|x\<rightarrow\>0<rsup|->>
        <frac|1|x>=-\<infty\>.
      </equation*>
    </example*>

    <\example*>
      \;

      <\equation*>
        lim<rsub|x\<rightarrow\>0<rsup|+>> ln
        x=-\<infty\>,<space|1em>lim<rsub|x\<rightarrow\><around*|(|<frac|\<pi\>|2>|)><rsup|->>
        tan x=\<infty\>,<space|1em><space|1em>lim<rsub|x\<rightarrow\><around*|(|<frac|\<pi\>|2>|)><rsup|+>>
        tan x=-\<infty\>
      </equation*>
    </example*>

    <\definition>
      If the left/right hand limit of <math|f<around*|(|x|)>> is
      <math|\<infty\>> or <math|-\<infty\>> as <math|x> approaches <math|a>,
      then the line <math|x=a> is called a vertical asymptote of <math|f>.\ 
    </definition>

    \;

    <section|Calculating Limits Using the Limit Laws>

    <\wide-block>
      <tformat|<cwith|1|-1|1|-1|cell-lsep|2spc>|<cwith|1|-1|1|-1|cell-rsep|2spc>|<cwith|1|-1|1|-1|cell-bsep|2spc>|<cwith|1|-1|1|-1|cell-tsep|2spc>|<table|<row|<\cell>
        <strong|Limit Laws >(for limit or one-sided limits)

        <\enumerate>
          <item>Sum Law: <math|<with|math-display|true|lim<rsub|x\<rightarrow\>a>
          <around*|[|f<around*|(|x|)>+g<around*|(|x|)>|]>=lim<rsub|x\<rightarrow\>a>
          f<around*|(|x|)>+lim<rsub|x\<rightarrow\>a> g<around*|(|x|)>>>
          provided the limits on the right-hand-side exist.\ 

          <item>Difference Law: <math|<with|math-display|true|lim<rsub|x\<rightarrow\>a>
          <around*|[|f<around*|(|x|)>-g<around*|(|x|)>|]>=lim<rsub|x\<rightarrow\>a>
          f<around*|(|x|)>-lim<rsub|x\<rightarrow\>a> g<around*|(|x|)>>>
          provided the limits on the right-hand-side exist.\ 

          <item>Constant Multiple Law: <math|<with|math-display|true|lim<rsub|x\<rightarrow\>a>
          c*f<around*|(|x|)>=c*lim<rsub|x\<rightarrow\>a>> f<around*|(|x|)>>
          provided the limit on the right-hand-side exists, where <math|c> is
          a constant. \ 

          <item>Product Law: <math|<with|math-display|true|lim<rsub|x\<rightarrow\>a>
          <around*|[|f<around*|(|x|)>*g<around*|(|x|)>|]>=lim<rsub|x\<rightarrow\>a>
          f<around*|(|x|)>*lim<rsub|x\<rightarrow\>a> g<around*|(|x|)>>>
          provided the limits on the right-hand-side exist.\ 

          <item>Quotient Law: \ <math|<with|math-display|true|lim<rsub|x\<rightarrow\>a>
          <frac|f<around*|(|x|)>|g<around*|(|x|)>>=<frac|<with|math-display|true|lim<rsub|x\<rightarrow\>a>
          f<around*|(|x|)>>|<with|math-display|true|lim<rsub|x\<rightarrow\>a>
          g<around*|(|x|)>>>>> provided the limits on the right-hand-side
          exist and <math|lim<rsub|x\<rightarrow\>a>
          g<around*|(|x|)>\<neq\>0>.\ 
        </enumerate>
      </cell>>>>
    </wide-block>

    \;

    Generalize to any <strong|algebraic combination> of functions: e.g.\ 

    <\equation*>
      lim<rsub|x\<rightarrow\>a> 2*f<around*|(|x|)>+5*g<around*|(|x|)>=2*lim<rsub|x\<rightarrow\>a>
      f<around*|(|x|)>+5*lim<rsub|x\<rightarrow\>a> g<around*|(|x|)>.
    </equation*>

    Power Law: <math|lim<rsub|x\<rightarrow\>a>
    f<around*|(|x|)><rsup|n>=<around*|[|lim<rsub|x\<rightarrow\>a>
    f<around*|(|x|)>|]><rsup|n>>.\ 

    Root Law: <math|lim<rsub|x\<rightarrow\>a>
    f<around*|(|x|)><rsup|1/n>=<around*|[|lim<rsub|x\<rightarrow\>a>
    f<around*|(|x|)>|]><rsup|1/n>> (<math|n> must be odd if
    <math|lim<rsub|x\<rightarrow\>a> f<around*|(|x|)>\<less\>0>).\ 

    Two simple limits:

    <\equation*>
      lim<rsub|x\<rightarrow\>a> c=c,<space|1em>lim<rsub|x\<rightarrow\>a>
      x=a.
    </equation*>

    \;

    <\wide-block>
      <tformat|<cwith|1|1|1|1|cell-lsep|1spc>|<cwith|1|1|1|1|cell-rsep|1spc>|<cwith|1|1|1|1|cell-bsep|1spc>|<cwith|1|1|1|1|cell-tsep|1spc>|<table|<row|<\cell>
        <strong|Direct substitution property> for polynomials and rational
        functions:

        <\equation*>
          lim<rsub|x\<rightarrow\>a> P<around*|(|x|)>=P<around*|(|a|)>,<space|1em>lim<rsub|x\<rightarrow\>a>
          <frac|P<around*|(|x|)>|Q<around*|(|x|)>>=<frac|P<around*|(|a|)>|Q<around*|(|a|)>>,<text|if
          >Q<around*|(|a|)>\<neq\>0.
        </equation*>

        where <math|P,Q> are polynomials.\ 
      </cell>>>>
    </wide-block>

    \;

    <\wide-block>
      <tformat|<cwith|1|-1|1|-1|cell-lsep|1spc>|<cwith|1|-1|1|-1|cell-rsep|1spc>|<cwith|1|-1|1|-1|cell-bsep|1spc>|<cwith|1|-1|1|-1|cell-tsep|1spc>|<table|<row|<\cell>
        <\example>
          \;

          <\equation*>
            lim<rsub|x\<rightarrow\>2> <around*|(|x<rsup|3>-4x+5|)>=2<rsup|3>-4\<cdot\>2+5=5.
          </equation*>

          <\equation*>
            lim<rsub|x\<rightarrow\>1> <frac|x<rsup|2>-2x|x<rsup|5>+1>=<frac|1-2|2>=-<frac|1|2>.
          </equation*>

          <\equation*>
            lim<rsub|x\<rightarrow\>1> <frac|x<rsup|2>-2x|x<rsup|3>-1>=lim<rsub|x\<rightarrow\>1>
            <frac|x<around*|(|x-2|)>|<around*|(|x-1|)><around*|(|x<rsup|2>+x+1|)>>
            <text|<space|1em>does not exist>
          </equation*>

          In fact

          <\equation*>
            lim<rsub|x\<rightarrow\>1<rsup|->>
            <frac|x<around*|(|x-2|)>|<around*|(|x-1|)><around*|(|x<rsup|2>+x+1|)>>=\<infty\>,<space|1em>lim<rsub|x\<rightarrow\>1<rsup|+>>
            <frac|x<around*|(|x-2|)>|<around*|(|x-1|)><around*|(|x<rsup|2>+x+1|)>>=-\<infty\>
          </equation*>

          Another example:

          <\equation*>
            lim<rsub|x\<rightarrow\>1> <frac|x<rsup|2>-x|x<rsup|3>-1>=lim<rsub|x\<rightarrow\>1><frac|x<around*|(|x-1|)>|<around*|(|x-1|)><around*|(|x<rsup|2>+x+1|)>>=lim<rsub|x\<rightarrow\>1><frac|x|<around*|(|x<rsup|2>+x+1|)>>=<frac|1|3>.
          </equation*>
        </example>
      </cell>>>>
    </wide-block>

    \;

    <\wide-block>
      <tformat|<cwith|1|1|1|1|cell-lsep|1spc>|<cwith|1|1|1|1|cell-rsep|1spc>|<cwith|1|1|1|1|cell-bsep|1spc>|<cwith|1|1|1|1|cell-tsep|1spc>|<table|<row|<\cell>
        If <math|f<around|(|x|)>=g<around|(|x|)>> when <math|x\<neq\>a>, then
        <math|lim<rsub|x\<rightarrow\>a> f<around|(|x|)>=lim<rsub|x\<rightarrow\>a>
        g<around|(|x|)>>, provided the limits exist.
      </cell>>>>
    </wide-block>

    \;

    <\theorem>
      If <math|f<around*|(|x|)>\<leqslant\>g<around*|(|x|)>> when <math|x> is
      near <math|a> (not including <math|a> itself), then

      <\equation*>
        lim<rsub|x\<rightarrow\>a> f<around*|(|x|)>\<leqslant\>lim<rsub|x\<rightarrow\>a>
        g<around*|(|x|)>
      </equation*>

      provided both limits exist.\ 
    </theorem>

    \;

    <\with|theorem-text|<macro|<localize|Theorem> (The Squeeze Theorem)>>
      <\theorem*>
        If <math|f<around*|(|x|)>\<leqslant\>g<around*|(|x|)>\<leqslant\>h<around*|(|x|)>>
        when <math|x> is near <math|a> (not including <math|a> itself), and

        <\equation*>
          lim<rsub|x\<rightarrow\>a> f<around*|(|x|)>=lim<rsub|x\<rightarrow\>a>
          h<around*|(|x|)>=L,
        </equation*>

        then

        <\equation*>
          lim<rsub|x\<rightarrow\>a> g<around*|(|x|)>=L.
        </equation*>

        Note: the theorem also holds for one-sided limits.\ 
      </theorem*>
    </with>

    <\wide-centered>
      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    </wide-centered>

    <\example*>
      <\equation*>
        lim<rsub|x\<rightarrow\>0> x*sin <frac|1|x>=?
      </equation*>

      For simplicity, first consider the case <math|x\<gtr\>0>. Then

      <\equation*>
        -x\<leqslant\> x*sin <frac|1|x>\<leqslant\>x,
      </equation*>

      and

      <\equation*>
        lim<rsub|x\<rightarrow\>0<rsup|+>>-x=lim<rsub|x\<rightarrow\>0<rsup|+>>
        x=0.
      </equation*>

      By the squeeze theorem, we know

      <\equation*>
        lim<rsub|x\<rightarrow\>0<rsup|+>> x*sin <frac|1|x>=0.
      </equation*>

      Similarly, we have

      <\equation*>
        lim<rsub|x\<rightarrow\>0<rsup|->> x*sin <frac|1|x>=0.
      </equation*>

      So

      <\equation*>
        lim<rsub|x\<rightarrow\>0> x*sin <frac|1|x>=0.
      </equation*>
    </example*>

    \;

    <section|(skip)>

    <section|Continuity>

    <\definition>
      A function <math|f> is said to be <strong|continuous> at <math|a>, if

      <\equation*>
        lim<rsub|x\<rightarrow\>a> f<around*|(|x|)>=f<around*|(|a|)>.
      </equation*>

      If <math|f> is defined near <math|a> and <math|f> is not continuous at
      <math|a>, we say <math|f> is <strong|discontinuous> at <math|a>.\ 
    </definition>

    \;

    <\wide-block>
      <tformat|<cwith|1|-1|1|-1|cell-lsep|2spc>|<cwith|1|-1|1|-1|cell-rsep|2spc>|<cwith|1|-1|1|-1|cell-bsep|2spc>|<cwith|1|-1|1|-1|cell-tsep|2spc>|<table|<row|<\cell>
        <\example>
          At which numbers is <math|f> continuous?

          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6285A4D390F6A3CF95422B4D2AF1F4B59F68547DA466A2CE2A89E608C29447848E2008A74E9DC8D055FBC0712193F8578F7FE5914079E0256F5127AF5344D00882700244E80882F081A026F8EC641F2108823059488D8E20088220080D8B081D41100441101A16113A8220088220342C22740441100441685844E80882200882D0B088D01104411004A16111A1230882200842C322424710044110848645848E20088220080D8B081D41100441101A16113A8220088220342C22740441100441685844E80882200882D0B088D01104411004A16111A1230882200842C32242E77DB0D6A2943AEEF6896E1304411004E1F44284CE09B0D6A2B5269D4E53A954E2DB9552586B09C3106BED87F8130A82200882F07E88D0790FACB5789E47B95C062291A3B5C618833146848E200882209CE6F81FF60F703AE2795E1CB9191E1EC6F3BCF87B4EE4F8BE2FE92B4110044138CD9188CE0424EB723A3B3B696A6A02200C439452F8BEE843411004413813902BF6041863E2BA9C42A110A7A84E54982C08822008C2E98944744E80310680E1E1610A8502409CC272DF1304411004E1F446223A13E0FB3E5D5D5DF4F4F490C964E254958BE638A1A3B5E84441100441389D11A13301BEEFB372E54ABEFCE52F93C964E8EDED05A21A1DCFF3F03C8F200800113B82200882703AA300ACF449D7906C1F77C5C7E57219630C994C06A5146118A2B5969A1D41100441388D11A1F33E24058F31268EE438C12308822008C2E98BA4AEDE036B6D2C72EAA337227204411004E1F447223A8220088220342C52492B0882200842C322424710044110848645848E20088220080D8B081D41100441101A16113A8220088220342C22740441100441685844E80882200882D0B0FC3FC62EDED59CD670A90000000049454E44AE426082>|png>|0.4par|||>
        </example>
      </cell>|<\cell>
        <\itemize>
          <item><math|f> is discontinuous at <math|1> since
          <math|f<around*|(|1|)>> is undefined

          <item><math|f> is discontinuous at <math|3> since
          <math|lim<rsub|x\<rightarrow\>3> f<around*|(|x|)>> does not exist.\ 

          <item><math|f> is discontinuous at <math|5> since
          <math|lim<rsub|x\<rightarrow\>5>
          f<around*|(|x|)>\<neq\>f<around*|(|5|)>>.\ 

          <item><math|f> is continuous at other points.\ 
        </itemize>
      </cell>>>>
    </wide-block>

    \;

    <\note>
      A polynomial <math|P<around*|(|x|)>> is continuous for every
      <math|x\<in\><with|font|Bbb|R>>.\ 
    </note>

    <\example>
      Where are each of the following functions discontinuous?

      <\enumerate-alpha>
        <item><math|f<around|(|x|)>=<dfrac|x<rsup|2>-x-2|x-2>>

        <\itemize>
          <item>If <math|x\<neq\>2>, then <math|f> is continuous at <math|x.>

          <item>If <math|x=2>, then <math|f> is discontinuous at <math|x>,
          since <math|f<around*|(|2|)>> is undefined. However,\ 

          <\equation*>
            lim<rsub|x\<rightarrow\>2> f<around|(|x|)>=lim<rsub|x\<rightarrow\>2>
            <dfrac|x<rsup|2>-x-2|x-2>=lim<rsub|x\<rightarrow\>2>
            <dfrac|<around*|(|x+1|)>*<around*|(|x-2|)>|x-2>=lim<rsub|x\<rightarrow\>2>
            <around*|(|x+1|)>=3.
          </equation*>

          If we define

          <\equation*>
            <wide|f|~><around*|(|x|)>=<choice|<tformat|<table|<row|<cell|<dfrac|x<rsup|2>-x-2|x-2>,>|<cell|x\<neq\>2>>|<row|<cell|3,>|<cell|x=2>>>>>
          </equation*>

          Then <math|f> is continuous everywhere (including <math|x=2>). Thus
          this kind of discontinuity is called <strong|removable>.\ 
        </itemize>

        <item><math|f<around|(|x|)>=<around*|{|<tabular*|<tformat|<cwith|1|-1|1|1|cell-halign|l>|<cwith|1|-1|1|1|cell-lborder|0ln>|<cwith|1|-1|2|2|cell-halign|l>|<cwith|1|-1|3|3|cell-halign|l>|<cwith|1|-1|3|3|cell-rborder|0ln>|<table|<row|<cell|<dfrac|1|x<rsup|2>>>|<cell|<text|if
        >>|<cell|x\<neq\>0>>|<row|<cell|1>|<cell|<text|if
        >>|<cell|x=0>>>>>|\<nobracket\>>>,\ 

        <\equation*>
          lim<rsub|x\<rightarrow\>0> f<around*|(|x|)>=\<infty\>
        </equation*>

        So the limit does not exist, and <math|f> can not be continuous at
        <math|x=0>. This type of discontinuity is called <strong|infinite
        discontinuity>.\ 

        <item><math|f<around|(|x|)>=<around*|{|<tabular*|<tformat|<cwith|1|-1|1|1|cell-halign|l>|<cwith|1|-1|1|1|cell-lborder|0ln>|<cwith|1|-1|2|2|cell-halign|l>|<cwith|1|-1|3|3|cell-halign|l>|<cwith|1|-1|3|3|cell-rborder|0ln>|<table|<row|<cell|<dfrac|x<rsup|2>-x-2|x-2>>|<cell|<text|if
        >>|<cell|x\<neq\>2>>|<row|<cell|1>|<cell|<text|if
        >>|<cell|x=2>>>>>|\<nobracket\>>>

        Similar to part (a), <math|f> is discontinuous at <math|x=2> since\ 

        <\equation*>
          lim<rsub|x\<rightarrow\>2> f<around*|(|x|)>=3\<neq\>f<around*|(|2|)>
        </equation*>

        This is still a removable discontinuity.\ 

        <item><math|f<around*|(|x|)>=<around*|\<llbracket\>|x|\<rrbracket\>>>,
        the greatest integer smaller than or equal to <math|x>. For example

        <\equation*>
          <around*|\<llbracket\>|1.3|\<rrbracket\>>=1,<space|1em><around*|\<llbracket\>|1|\<rrbracket\>>=1,<space|1em><around*|\<llbracket\>|-2.7|\<rrbracket\>>=-3.
        </equation*>

        <math|f> is continuous everywhere except when
        <math|x\<in\>\<bbb-Z\>>: the set of all integers. The reason is

        <\equation*>
          lim<rsub|x\<rightarrow\>n<rsup|->>
          f<around*|(|x|)>=n-1,<space|1em>lim<rsub|x\<rightarrow\>n<rsup|+>>
          f<around*|(|x|)>=n<space|1em>\<Rightarrow\><space|1em>lim<rsub|x\<rightarrow\>n>
          f<around*|(|x|)>*<text|<space|1em>does not exist>
        </equation*>

        where <math|n\<in\>\<bbb-Z\>>. So there is a jump at <math|x=n>. We
        call it a <strong|jump discontinuity>.\ 
      </enumerate-alpha>

      \;

      There are other cases of discontinuity. For example

      <\equation*>
        f<around*|(|x|)>=sin <frac|1|x>
      </equation*>

      <math|f<around*|(|x|)>> has no left or right hand limit as
      <math|x\<rightarrow\>0>.\ 
    </example>

    <\definition>
      A function <math|f<around*|(|x|)>> is said to be <strong|continuous
      from the left> at <math|x=a> if

      <\equation*>
        lim<rsub|x\<rightarrow\>a<rsup|->> f<around*|(|x|)>=f<around*|(|a|)>.
      </equation*>

      Similarly we can define <strong|continuous from the right>.\ 
    </definition>

    \;

    For example, the function <math|f<around*|(|x|)>=<around*|\<llbracket\>|x|\<rrbracket\>>>
    is continuous from the right at <math|x=n\<in\>\<bbb-Z\>>.\ 

    \;

    <\definition>
      A function <math|f<around*|(|x|)>> is said to be <strong|continuous on
      an interval> <math|I> if it's continuous at every point
      <math|x\<in\>I>. Here <math|I> can be all types of intervals, such as

      <\equation*>
        <around*|(|a,b|)>,<around*|[|a,b|]>,<around*|[|a,b|)>,<around*|[|a,\<infty\>|)>,<around*|(|-\<infty\>,a|)>*<text|<space|1em>etc.>
      </equation*>
    </definition>

    \;

    <\theorem>
      If <math|f> and <math|g> are both continuous at <math|x=a>. Then

      <\equation*>
        f+g,f-g,f*g,c*f
      </equation*>

      are continuous at <math|x=a>, where <math|c> is a constant. Moreover,\ 

      <\equation*>
        <frac|f|g>
      </equation*>

      is continuous at <math|x=a> provided <math|g<around*|(|a|)>\<neq\>0>.
    </theorem>

    \;

    <\theorem>
      A polynomial <math|P<around*|(|x|)>> is continuous for all <math|x>. A
      rational function <math|P<around*|(|x|)>/Q<around*|(|x|)>> is
      continuous in its domain.\ 
    </theorem>

    \;

    \;

    <strong|Trigonometric functions>

    From the geometrical definition of <math|sin x>, we know

    <\equation*>
      lim<rsub|x\<rightarrow\>0> sin x=0=sin 0
    </equation*>

    So <math|sin x> is continuous at <math|0>. Similarly we can show
    <math|cos*x> is continuous at <math|0>.\ 

    How about any <math|a>?

    <\equation*>
      sin <around*|(|a+x|)>=<around*|(|sin a|)>*<around*|(|cos
      x|)>+<around*|(|cos a|)>*<around*|(|sin x|)>\<rightarrow\>sin
      a<space|1em><text|as><space|1em>x\<rightarrow\>0
    </equation*>

    So <math|sin x> is continuous at every <math|a>. Similarly we can show
    <math|cos x> is continuous at every <math|a>. From the quotient rule, \ 

    <\equation*>
      tan x=<frac|sin x|cos x>
    </equation*>

    is continuous in its domain, i.e. every <math|x> except
    <math|x=<frac|2*k+1|2>*\<pi\>> for <math|k\<in\>\<bbb-Z\>>.\ 

    <\question>
      What type of discontinuity does <math|tan x> have at
      \ <math|x=<frac|2*k+1|2>*\<pi\>> for <math|k\<in\>\<bbb-Z\>>?
    </question>

    Similarly, we deduce that all the trigonometric and inverse trigonometric
    functions are continuous in their domains.\ 

    \;

    Other functions: root functions, exponential functions, logarithmic
    functions are continuous at every number in the domains.\ 

    <\theorem>
      If <math|f> is continuous at <math|b> and
      <math|lim<rsub|x\<rightarrow\>a> g<around*|(|x|)>=b>, then
      <math|lim<rsub|x\<rightarrow\>a> f<around*|(|g<around*|(|x|)>|)>=f<around*|(|b|)>>,
      i.e.\ 

      <\equation*>
        lim<rsub|x\<rightarrow\>a> f<around*|(|g<around*|(|x|)>|)>=f<around*|(|lim<rsub|x\<rightarrow\>a>
        g<around*|(|x|)>|)>.
      </equation*>
    </theorem>

    \;

    <\theorem>
      If <math|g> is continuous at <math|a> and <math|f> is continuous at
      <math|g<around*|(|a|)>>, then <math|f\<circ\>g> is continuous at
      <math|a>.\ 
    </theorem>

    \;

    <\example>
      The function

      <\equation*>
        f<around*|(|x|)>=sin <around*|(|1+x<rsup|2>|)>
      </equation*>

      is continuous at every number.\ 
    </example>

    <\example>
      The function

      <\equation*>
        f<around*|(|x|)>=<frac|1+2<rsup|x>+sin x|<sqrt|1-cos x>>
      </equation*>

      is continuous everywhere except where <math|cos x=1>, i.e.
      <math|x=2*k*\<pi\>> for <math|k\<in\>\<bbb-Z\>>.\ 
    </example>

    <\render-theorem|<strong|Intermediate Value Theorem>>
      Suppose that <math|f> is continuous on the closed interval
      <math|<around|[|a,b|]>> and let <math|m> be any number between
      <math|f<around|(|a|)>> and <math|f<around|(|b|)>>, where
      <math|f<around|(|a|)>\<neq\>f<around|(|b|)>>. Then there exists a
      number <math|c> in <math|<around|(|a,b|)>> such that
      <math|f<around|(|c|)>=m>.
    </render-theorem>

    \;

    \;

    Example. Show that the equation <math|x<rsup|2>-x-1=<frac|1|x+1>> has a
    solution for <math|1\<less\>x\<less\>2>.

    <strong|Answer:> Let

    <\equation*>
      f<around*|(|x|)>=x<rsup|2>-x-1-<frac|1|x+1>
    </equation*>

    We want to find <math|x> such that <math|f<around*|(|x|)>=0>. Now

    <\equation*>
      f<around*|(|1|)>\<less\>0,<space|1em>f<around*|(|2|)>\<gtr\>0
    </equation*>

    So <math|0> is a number between <math|f<around*|(|1|)>> and
    <math|f<around*|(|2|)>>. Moreover, <math|f> is continuous in
    <math|<around*|[|1,2|]>>. \ By the intermediate value theorem, there
    exists an <math|x\<in\><around*|(|1,2|)>> such that
    <math|f<around*|(|x|)>=0>.\ 

    We can now pick the midpoint of <math|1> and <math|2>, that is
    <math|<frac|3|2>>, then

    <\equation*>
      f<around*|(|<frac|3|2>|)>=<frac|9|4>-<frac|3|2>-1-<frac|2|5>\<less\>0.
    </equation*>

    Using the Intermediate value theorem again in
    <math|<around*|[|<frac|3|2>,2|]>>, we know there exists an
    <math|x\<in\><around*|(|<frac|3|2>,2|)>> such that
    <math|f<around*|(|x|)>=0>.\ 

    \;

    Repeating this procedure, we can get more and more accurate
    approximations of the root of <math|f<around*|(|x|)>>. This is called the
    <strong|bisection method>.

    <section|Limits at Infinity; Horizontal Asymptotes>

    motivation: <math|lim<rsub|x\<rightarrow\>\<infty\>>
    <frac|x<rsup|2>-1|x<rsup|2>+1>>

    <\equation*>
      lim<rsub|x\<rightarrow\>\<infty\>> <frac|x<rsup|2>-1|x<rsup|2>+1>=lim<rsub|x\<rightarrow\>\<infty\>><frac|1-<frac|1|x<rsup|2>>|1+<frac|1|x<rsup|2>>>=<frac|1-0|1+0>=1
    </equation*>

    <\definition>
      Suppose <math|f<around*|(|x|)>> is defined for <math|x\<gtr\>a> for
      some <math|a>. Then we define

      <\equation*>
        lim<rsub|x\<rightarrow\>\<infty\>> f<around*|(|x|)>=L
      </equation*>

      if <math|f<around*|(|x|)>> can be <strong|arbitrarily> close to
      <math|L> as long as <math|x> is <strong|sufficiently> large. Similarly
      we can define

      <\equation*>
        lim<rsub|x\<rightarrow\>-\<infty\>> f<around*|(|x|)>=L
      </equation*>
    </definition>

    \;

    <\definition>
      If

      <\equation*>
        lim<rsub|x\<rightarrow\>\<infty\>>
        f<around*|(|x|)>=L<space|1em>or<space|1em>lim<rsub|x\<rightarrow\>-\<infty\>>
        f<around*|(|x|)>=L,
      </equation*>

      we call <math|y=L> a <strong|horizontal asymptote> of <math|f>.
    </definition>

    \;

    <\example>
      Let <math|f<around*|(|x|)>=<frac|1|x>>. Then
      <math|lim<rsub|x\<rightarrow\>\<infty\>> f<around*|(|x|)>=0>. So
      <math|y=0> is horizontal asymptote of <math|f>. Moreover,
      <math|lim<rsub|x\<rightarrow\>0<rsup|+>> f<around*|(|x|)>=\<infty\>>,
      so <math|x=0> is a vertical asymptote of <math|f>.
    </example>

    <\example>
      \ 

      <\equation*>
        lim<rsub|x\<rightarrow\>\<infty\>>
        <frac|3*x<rsup|2>-x-2|5*x<rsup|2>+4*x+1>=lim<rsub|x\<rightarrow\>\<infty\>>
        <frac|3*-<frac|1|x>-<frac|2|x<rsup|2>>|5*+<frac|4|x>+<frac|1|x<rsup|2>>>=<frac|3|5>.
      </equation*>
    </example>

    <\example>
      \ 

      <\equation*>
        lim<rsub|x\<rightarrow\>\<infty\>>
        <frac|<sqrt|2*x<rsup|2>+1>|3*x*-5>=lim<rsub|x\<rightarrow\>\<infty\>>
        <frac|<sqrt|2*+<frac|1|x<rsup|2>>>|3**-<frac|5|x>>=<frac|<sqrt|2>|3>
      </equation*>

      <\equation*>
        lim<rsub|x\<rightarrow\>-\<infty\>>
        <frac|<sqrt|2*x<rsup|2>+1>|3*x*-5>=lim<rsub|x\<rightarrow\>-\<infty\>>
        <frac|<frac|1|x><sqrt|2*x<rsup|2>+1>|3**-<frac|5|x>>=lim<rsub|x\<rightarrow\>-\<infty\>>
        <frac|-<sqrt|2*+<frac|1|x<rsup|2>>>|3**-<frac|5|x>>=-<frac|<sqrt|2>|3>
      </equation*>

      So we have two horizontal asymptote <math|y=<sqrt|2>/3> and
      <math|y=-<sqrt|2>/3>. Moreover

      <\equation*>
        lim<rsub|x\<rightarrow\><frac|5|3><rsup|+>>
        <frac|<sqrt|2*x<rsup|2>+1>|3*x*-5>=\<infty\>,<space|1em>lim<rsub|x\<rightarrow\><frac|5|3><rsup|->>
        <frac|<sqrt|2*x<rsup|2>+1>|3*x*-5>=-\<infty\>.
      </equation*>

      So <math|x=5/3> is a vertical asymptote.
    </example>

    <unfolded|<\question>
      <math|lim<rsub|x\<rightarrow\>\<infty\>> sin <frac|1|x>=?>
    </question>|<\answer*>
      <\equation*>
        lim<rsub|x\<rightarrow\>\<infty\>> sin
        <frac|1|x>=lim<rsub|t\<rightarrow\>0<rsup|+>> sin t=sin 0=0.
      </equation*>
    </answer*>>

    <unfolded|<\question>
      <math|lim<rsub|x\<rightarrow\>\<infty\>> x*<around*|(|sin
      <frac|1|x>|)>=?>
    </question>|<\answer*>
      <\equation*>
        lim<rsub|x\<rightarrow\>\<infty\>> x*<around*|(|sin
        <frac|1|x>|)>=lim<rsub|x\<rightarrow\>\<infty\>> *<frac|sin
        <frac|1|x>|<frac|1|x>>=lim<rsub|t\<rightarrow\>0<rsup|+>> <frac|sin
        t|t>=1
      </equation*>
    </answer*>>

    <\definition>
      Suppose <math|f> is defined at every <math|x\<gtr\>a> for some
      <math|a>. Then we define

      <\equation*>
        lim<rsub|x\<rightarrow\>\<infty\>> f<around*|(|x|)>=\<infty\>
      </equation*>

      if <math|f<around*|(|x|)>> can be arbitrarily large as long as <math|x>
      is sufficiently large. Similarly we can define

      <\equation*>
        lim<rsub|x\<rightarrow\>\<infty\>>
        f<around*|(|x|)>=-\<infty\>,<space|1em>lim<rsub|x\<rightarrow\>-\<infty\>>
        f<around*|(|x|)>=\<infty\>,<space|1em>lim<rsub|x\<rightarrow\>-\<infty\>>
        f<around*|(|x|)>=-\<infty\>
      </equation*>
    </definition>

    <\example>
      \ 

      <\equation*>
        lim<rsub|x\<rightarrow\>\<infty\>>
        x<rsup|2>=\<infty\>,<space|1em>lim<rsub|x\<rightarrow\>-\<infty\>>
        x<rsup|2>=\<infty\>,<space|1em>lim<rsub|x\<rightarrow\>-\<infty\>>
        x<rsup|3>=-\<infty\>.
      </equation*>
    </example>

    <\note>
      If <math|P<around*|(|x|)>> is a polynomial, then

      <\equation*>
        lim<rsub|x\<rightarrow\>\<infty\>>
        P<around*|(|x|)>=\<pm\>\<infty\>,<space|1em>lim<rsub|x\<rightarrow\>-\<infty\>>
        P<around*|(|x|)>=\<pm\>\<infty\>.
      </equation*>
    </note>

    <\example>
      A business manager determines that the total cost of producing <math|x>
      units of a particular commodity may be modeled by the function

      <\equation*>
        C<around|(|x|)>=7.5*x+120,000
      </equation*>

      The average cost is <math|A<around|(|x|)>=<frac|C<around|(|x|)>|x>>.
      Find <math|lim<rsub|x\<rightarrow\>+\<infty\>> A<around|(|x|)>> and
      interpret your result.
    </example>

    <strong|Answer:>\ 

    <\equation*>
      lim<rsub|x\<rightarrow\>\<infty\>> <frac|7.5*x+120000|x>=7.5
    </equation*>

    <gap-wide|>

    <section|Derivatives and Rates of Change>

    tangent line, velocity, derivative, rates of change

    \;

    The slope of the secant line for a function <math|y=f<around*|(|x|)>>
    from <math|x=a> to <math|x=b> is

    <\equation*>
      m=<frac|f<around*|(|b|)>-f<around*|(|a|)>|b-a>
    </equation*>

    If <math|b\<rightarrow\>a>, then the slope approaches the slope of the
    tangent line if it exists at <math|x=a>.

    <\definition>
      If <math|f> is continuous at <math|a>, then we define

      <\equation*>
        f<rprime|'><around*|(|a|)>=lim<rsub|x\<rightarrow\>a>
        <frac|f<around*|(|x|)>-f<around*|(|a|)>|x-a>
      </equation*>

      to be the <strong|derivative> of <math|f> at <math|a>. If <math|f> has
      a derivative at <math|a>, we say <math|f> is <strong|differentiable> at
      <math|a>.
    </definition>

    \;

    \;

    Other motivations: velocity.\ 

    Suppose <math|f<around*|(|t|)>> is the displacement of an object at time
    <math|t>. Then the <strong|average velocity> from <math|t=a> to
    <math|t=b> is

    <\equation*>
      <frac|f<around*|(|b|)>-f<around*|(|a|)>|b-a>.
    </equation*>

    The <strong|instantaneous velocity> at <math|t=a> is

    <\equation*>
      v=lim<rsub|x\<rightarrow\>a> <frac|f<around*|(|x|)>-f<around*|(|a|)>|x-a>=f<rprime|'><around*|(|a|)>.
    </equation*>

    Similarly, the acceleration is the derivative of velocity:
    <math|a=v<rprime|'>=f<rprime|''>>.

    \;

    Generally speaking, <math|y=f<rprime|'><around*|(|a|)>> is the
    <strong|rate of change> of <math|y> with respect to <math|t> at <math|a>.

    \;

    <\example>
      Find where the function <math|f<around*|(|x|)>=<around*|\||x|\|>> is
      differentiable and find its derivative.

      <\wide-centered>
        <with|gr-mode|<tuple|edit|point>|gr-frame|<tuple|scale|1cm|<tuple|0.5gw|0.5gh>>|gr-geometry|<tuple|geometry|0.4par|0.6par|center>|gr-grid|<tuple|empty>|gr-edit-grid-aspect|<tuple|<tuple|axes|none>|<tuple|1|none>|<tuple|10|none>>|gr-edit-grid|<tuple|empty>|gr-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-color|blue|gr-auto-crop|true|<graphics||<with|magnify|0.6187192033860641|arrow-end|\<gtr\>|<line|<point|-0.6976773750669972|-0.4725822239854681>|<point|3.064795878819595|-0.4280933012526935>>>|<with|magnify|0.6187192033860641|arrow-end|\<gtr\>|<line|<point|1.1962703665956091|-1.0191569120991069>|<point|1.1390705004840198|2.0505631263079476>>>|<with|magnify|0.6187192033860641|color|blue|<line|<point|-0.4561665212172813|1.1798577574234392>|<point|1.1856707656828998|-0.4503127952722712>>>|<with|magnify|0.6187192033860641|color|blue|<line|<point|1.1856707785628005|-0.4503126636979932>|<point|2.5944867105132174|1.2751885515325685>>>|<with|color|blue|<math-at|<with|color|blue|y=<around*|\||x|\|>>|<point|1.6265770454169626|1.0483157106310128>>>|<with|color|blue|<text-at|corner|<point|0.36684727099403713|-0.7703082594865909>>>|<with|color|blue|<point|1.18567|-0.450313>>>>
      </wide-centered>

      <\enumerate>
        <item>If <math|a\<gtr\>0>,

        <\equation*>
          lim<rsub|x\<rightarrow\>a> <frac|f<around*|(|x|)>-f<around*|(|a|)>|x-a>=lim<rsub|x\<rightarrow\>a>
          <frac|<around*|\||x|\|>-<around*|\||a|\|>|x-a>=lim<rsub|x\<rightarrow\>a>
          <frac|x-a|x-a>=1<space|1em>\<Rightarrow\><space|1em>f<rprime|'><around*|(|a|)>=1.
        </equation*>

        <item>If <math|a\<less\>0>,

        <\equation*>
          lim<rsub|x\<rightarrow\>a> <frac|f<around*|(|x|)>-f<around*|(|a|)>|x-a>=lim<rsub|x\<rightarrow\>a>
          <frac|<around*|\||x|\|>-<around*|\||a|\|>|x-a>=lim<rsub|x\<rightarrow\>a>
          <frac|-x+a|x-a>=-1<space|1em>\<Rightarrow\><space|1em>f<rprime|'><around*|(|a|)>=-1.
        </equation*>

        <item>If <math|a=0>,\ 

        <\equation*>
          lim<rsub|x\<rightarrow\>0> <frac|f<around*|(|x|)>-f<around*|(|0|)>|x>=lim<rsub|x\<rightarrow\>0>
          <frac|<around*|\||x|\|>|x> <text| does not
          exist><space|1em>\<Rightarrow\><space|1em>f<rprime|'><around*|(|0|)>
          <text| does not exist.>
        </equation*>
      </enumerate>
    </example>

    Other examples where the function is not differentiable.

    <\wide-centered>
      <with|gr-mode|<tuple|edit|text-at>|gr-frame|<tuple|scale|0.75cm|<tuple|0.5gw|0.5gh>>|gr-geometry|<tuple|geometry|1par|0.6par>|gr-grid|<tuple|empty>|gr-edit-grid-aspect|<tuple|<tuple|axes|none>|<tuple|1|none>|<tuple|10|none>>|gr-edit-grid|<tuple|empty>|gr-dash-style|11100|gr-color|red|gr-auto-crop|true|magnify|0.75|gr-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-old|<tuple|cartesian|<point|0|0>|1>|<graphics||<spline|<point|-5.4|0.1>|<point|-4.9|-0.7>|<point|-3.3|-0.2>>|<spline|<point|-3.3|0.8>|<point|-2.1000000000000005|1.6>|<point|-1.3000000000000003|1.5>>|<point|-3.3|-0.2>|<with|point-style|round|<point|-3.3|0.8>>|<spline|<point|0.3|0.5>|<point|1.8000000000000003|2.1>|<point|2.799999999999999|0.8>|<point|3.200000000000001|-1.0>|<point|3.8999999999999995|-0.8>>|<with|dash-style|11100|color|red|<line|<point|2.9|1.6>|<point|2.8999999999999995|-1.3>>>|<with|color|red|<point|2.9|-0.0339431>>|<with|color|red|<text-at|discontinuous|<point|-4|-1.2>>>|<with|color|red|<text-at|vertical
      tangent|<point|1.8000000000000003|-1.8>>>>>
    </wide-centered>

    \;

    \;

    <\question>
      Show that if <math|f<around*|(|a|)>> is defined and

      <\equation*>
        lim<rsub|x\<rightarrow\>a> <frac|f<around*|(|x|)>-f<around*|(|a|)>|x-a>
        <text|exist>,
      </equation*>

      then <math|f> is continuous at <math|a>.
    </question>

    <\summarized-plain>
      <section|The Derivative as a Function>

      definition, graph, other notations
    <|summarized-plain>
      \;
    </summarized-plain>

    <\math>
      <\equation*>
        f<rprime|'><around*|(|a|)>=lim<rsub|x\<rightarrow\>a>
        <frac|f<around*|(|x|)>-f<around*|(|a|)>|x-a>=lim<rsub|h\<rightarrow\>0>
        <frac|f<around*|(|a+h|)>-f<around*|(|a|)>|h>
      </equation*>
    </math>

    Now for a variable <math|x>, we obtain a function

    <\equation*>
      f<rprime|'><around*|(|x|)>=lim<rsub|h\<rightarrow\>0>
      <frac|f<around*|(|x+h|)>-f<around*|(|x|)>|h>
    </equation*>

    called the <strong|derivative function> of <math|f>, or simply the
    <strong|derivative> of <math|f>.

    <\example>
      Let <math|f<around*|(|x|)>=x<rsup|2>>, find
      <math|f<rprime|'><around*|(|x|)>>.

      \;

      <strong|Answer:>

      <\equation*>
        lim<rsub|h\<rightarrow\>0> <frac|f<around*|(|x+h|)>-f<around*|(|x|)>|h>=lim<rsub|h\<rightarrow\>0>
        <frac|<around*|(|x+h|)><rsup|2>-x<rsup|2>|h>=lim<rsub|h\<rightarrow\>0>
        <frac|2*x*h+h<rsup|2>|h>=lim<rsub|h\<rightarrow\>0>
        <around*|(|2*x+h|)>=2*x
      </equation*>

      So

      <\equation*>
        f<rprime|'><around*|(|x|)>=2*x,<space|1em>x\<in\><with|font|Bbb|R>.
      </equation*>
    </example>

    <\exercise>
      What is <math|f<rprime|'><around*|(|x|)>> if
      <math|f<around*|(|x|)>=x<rsup|n>,n\<in\>\<bbb-N\>>.
    </exercise>

    <\example>
      Let <math|f<around*|(|x|)>=<around*|\||x|\|>>, find
      <math|f<rprime|'><around*|(|x|)>>.

      \;

      <strong|Answer:>

      <\equation*>
        lim<rsub|h\<rightarrow\>0> <frac|f<around*|(|x+h|)>-f<around*|(|x|)>|h>=lim<rsub|h\<rightarrow\>0>
        <frac|<around*|\||x+h|\|>-<around*|\||x|\|>|h>.
      </equation*>

      Consider three cases

      <\enumerate>
        <item><math|x\<gtr\>0>

        <\equation*>
          lim<rsub|h\<rightarrow\>0> <frac|<around*|\||x+h|\|>-<around*|\||x|\|>|h>=lim<rsub|h\<rightarrow\>0>
          <frac|x+h-x|h>=1.
        </equation*>

        <item><math|x\<less\>0>

        <\equation*>
          lim<rsub|h\<rightarrow\>0> <frac|<around*|\||x+h|\|>-<around*|\||x|\|>|h>=lim<rsub|h\<rightarrow\>0>
          <frac|-<around*|(|x+h|)>+x|h>=-1.
        </equation*>

        <item><math|x=0.>

        <\equation*>
          lim<rsub|h\<rightarrow\>0> <frac|<around*|\||x+h|\|>-<around*|\||x|\|>|h>=lim<rsub|h\<rightarrow\>0>
          <frac|<around*|\||h|\|>|h> <text|does not exist>.
        </equation*>
      </enumerate>

      So

      <\equation*>
        f<rprime|'><around*|(|x|)>=<choice|<tformat|<table|<row|<cell|1,>|<cell|x\<gtr\>0,>>|<row|<cell|-1,>|<cell|x\<less\>0,>>|<row|<cell|<text|undefined,>>|<cell|x=0.>>>>>
      </equation*>
    </example>

    Given the graph of <math|f>, we can sketch the graph of
    <math|f<rprime|'>>.

    \;

    <\example>
      \;

      \;

      <\wide-tabular>
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        </cell>>>>
      </wide-tabular>
    </example>

    <\render-theorem>
      Other notations for derivative
    <|render-theorem>
      For a function <math|y=f<around*|(|x|)>>, the derivative function is

      <\equation*>
        f<rprime|'><around*|(|x|)>,<space|1em>y<rprime|'>,<space|1em><frac|d*y|d*x>,<space|1em>D<rsub|x>
        f,<space|1em>D f<space|1em>
      </equation*>

      The derivative at <math|a> is

      <\equation*>
        f<rprime|'><around*|(|a|)>,<space|1em><space|1em><around*|\<nobracket\>|<frac|d*y|d*x>|\|><rsub|x=a>,<space|1em>D<rsub|x>
        f<around*|(|a|)>,<space|1em>D f<around*|(|a|)>.
      </equation*>
    </render-theorem>

    \;

    \;

    \;

    \;

    \;

    \;

    \;
  </shown>>
</body>

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    <associate|auto-1|<tuple|?|1>>
    <associate|auto-10|<tuple|4|7>>
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<\auxiliary>
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    <\associate|toc>
      <vspace*|2fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|font-size|<quote|1.19>|Chapter
      2: Limits and Derivatives> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-1><vspace|1fn>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|1<space|2spc>The
      Tangent and Velocity Problems> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-2><vspace|0.5fn>

      <with|par-left|<quote|1tab>|1.1<space|2spc>The tangent problem (how to
      find the tangent line) <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-3>>

      <with|par-left|<quote|1tab>|1.2<space|2spc>The velocity problem (how to
      find instantaneous velocity) <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-4>>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|2<space|2spc>The
      Limit of a Function> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-5><vspace|0.5fn>

      <with|par-left|<quote|1tab>|2.1<space|2spc>Intuitive definition of a
      limit <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-6>>

      <with|par-left|<quote|1tab>|2.2<space|2spc>One-sided limits
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-7>>

      <with|par-left|<quote|1tab>|2.3<space|2spc>Infinite limits
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-8>>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|3<space|2spc>Calculating
      Limits Using the Limit Laws> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-9><vspace|0.5fn>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|4<space|2spc>(skip)>
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-10><vspace|0.5fn>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|5<space|2spc>Continuity>
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-11><vspace|0.5fn>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|6<space|2spc>Limits
      at Infinity; Horizontal Asymptotes>
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-12><vspace|0.5fn>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|7<space|2spc>Derivatives
      and Rates of Change> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-13><vspace|0.5fn>

      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|8<space|2spc>The
      Derivative as a Function> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-14><vspace|0.5fn>
    </associate>
  </collection>
</auxiliary>